JHI's elegant lower bound of $8$ on $N$ is achieved by an explicit dissection. I show my construction below; you might want to try to find a solution yourself before proceeding $-$ it makes for a neat puzzle. There may well be other ways to do it.

If somebody can make a "$3$-dimensional" graphic or picture of the $8$-piece dissection, you're welcome to add it by editing my answer. My diagrams are two-dimensional, labeling each piece with its height. Fortunately the dissection is simple enough for this to be possible; in particular, the eight pieces comprise four boxes and four L-shaped prisms. This also made it possible to find the solution using just pencil and paper on an otherwise uneventful international flight.

Begin by cutting the $6 \times 6 \times 6$ cube top to bottom into three pieces, as shown in top view in the first square diagram. Then cut each piece horizontally in two, dividing AB into $3+3$,$\phantom.$ C into $4+2$, and D into $5+1$. Each AB piece is then further subdivided into a box B and an L-shaped prism A. The second diagram shows (say) the bottom layer of four pieces, and the third diagram shows the top. Note that the AB subdivisions are not quite the same.

Pieces with the same color will come together to form a smaller cube. The larger C piece is a $4$-cube, and the two A pieces form a $3$-cube as shown. It remains to construct the $5$-cube from the remaining five pieces. The last two diagrams show the bottom and top of the $5$-cube.

The two $5$'s are the larger B piece, rotated to span the entire height of the cube, and the thick D piece. The thin D piece completes the bottom, with width $1$. The top is filled by the thinner C piece and the smaller B, both rotated to height 4. QEF

I guess that a physical model won't make for a hard puzzle to reconstitute into either one or three cubes (e.g. the AB, C, and D parts of the $6$-cube are independent) but would still make a nice physical model of the identity $3^3 + 4^3 + 5^3 = 6^3$.

This dissection is specific to the solution $(a,b,c;d)=(3,4,5;6)$ of the Diophantine equation $a^3+b^3+c^3=d^3$; I don't know whether an $8$-piece dissection is possible for any other solution. JHI's analysis shows that one can never get below $8$, and in some cases even that's not possible: if $a<b<c$ and $a+c<d$ then there's at least one corner of the $d$-cube, say $(1,1,1)$, that contributes to the $a$-cube, but then any cell $(x,y,z)$ with $\max(x,y,z) = a+1$ cannot connect to any corner. This first happens for $(a,b,c;d) = (6,32,33;41)$.

What's the minimal dissection for the "taxicab" identity $1^3 + 12^3 = 9^3 + 10^3$? JHI's corner-cutting argument shows that at least nine pieces are needed.

JHI's elegant lower bound of $8$ on $N$ is achieved by an explicit dissection. I show my construction below; you might want to try to find a solution yourself before proceeding $-$ it makes for a neat puzzle. There may well be other ways to do it.

If somebody can make a "$3$-dimensional" graphic or picture of the $8$-piece dissection, you're welcome to add it by editing my answer. My diagrams are two-dimensional, labeling each piece with its height. Fortunately the dissection is simple enough for this to be possible; in particular, the eight pieces comprise four boxes and four L-shaped prisms. This also made it possible to find the solution using just pencil and paper on an otherwise uneventful international flight.

Begin by cutting the $6 \times 6 \times 6$ cube top to bottom into three pieces, as shown in top view in the first square diagram. Then cut each piece horizontally in two, dividing AB into $3+3$,$\phantom.$ C into $4+2$, and D into $5+1$. Each AB piece is then further subdivided into a box B and an L-shaped prism A. The second diagram shows (say) the bottom layer of four pieces, and the third diagram shows the top. Note that the AB subdivisions are not quite the same.

^(source)

Pieces with the same color will come together to form a smaller cube. The larger C piece is a $4$-cube, and the two A pieces form a $3$-cube as shown. It remains to construct the $5$-cube from the remaining five pieces. The last two diagrams show the bottom and top of the $5$-cube.

^(source)

The two $5$'s are the larger B piece, rotated to span the entire height of the cube, and the thick D piece. The thin D piece completes the bottom, with width $1$. The top is filled by the thinner C piece and the smaller B, both rotated to height 4. QEF

I guess that a physical model won't make for a hard puzzle to reconstitute into either one or three cubes (e.g. the AB, C, and D parts of the $6$-cube are independent) but would still make a nice physical model of the identity $3^3 + 4^3 + 5^3 = 6^3$.

This dissection is specific to the solution $(a,b,c;d)=(3,4,5;6)$ of the Diophantine equation $a^3+b^3+c^3=d^3$; I don't know whether an $8$-piece dissection is possible for any other solution. JHI's analysis shows that one can never get below $8$, and in some cases even that's not possible: if $a<b<c$ and $a+c<d$ then there's at least one corner of the $d$-cube, say $(1,1,1)$, that contributes to the $a$-cube, but then any cell $(x,y,z)$ with $\max(x,y,z) = a+1$ cannot connect to any corner. This first happens for $(a,b,c;d) = (6,32,33;41)$.

What's the minimal dissection for the "taxicab" identity $1^3 + 12^3 = 9^3 + 10^3$? JHI's corner-cutting argument shows that at least nine pieces are needed.

JHI's elegant lower bound of $8$ on $N$ is achieved by an explicit dissection. I show my construction below; you might want to try to find a solution yourself before proceeding $-$ it makes for a neat puzzle. There may well be other ways to do it.

If somebody can make a "$3$-dimensional" graphic or picture of the dissection here, you're welcome to edit my answer to put it in. My diagrams are two-dimensional, labeling each piece with its height. Fortunately the dissection is simple enough for this to be possible; in particular, the eight pieces comprise four boxes and four L-shaped prisms. This also made it possible to find the solution using just pencil and paper on an otherwise uneventful international flight.

Begin by cutting the $6 \times 6 \times 6$ cube top to bottom into three pieces, as shown in top view the first square diagram. Then cut each piece horizontally in two, dividing AB into $3+3$,$\phantom.$ C into $4+2$, and D into $5+1$. Each AB piece is then further subdivided into a box B and an L-shaped prism A. The second diagram shows (say) the bottom layer of four pieces, and the third diagram shows the top. Note that the AB subdivisions are not quite the same.

Pieces with the same color will come together to form a smaller cube. The larger C piece is a $4$-cube, and the two A pieces form a $3$-cube as shown. It remains to construct the $5$-cube from the remaining five pieces. The last two diagrams show the bottom and top of the $5$-cube.

The two $5$'s are the larger B piece, rotated to span the entire height of the cube, and the thick D piece. The thin D piece completes the bottom, with width $1$. The top is filled by the thinner C piece and the smaller B, both rotated to height 4. QEF

I guess that a physical model won't be very hard to reconstitute into either one or three cubes (e.g. the AB, C, and D parts of the $6$-cube are independent) but would still make a nice physical model of the identity $3^3 + 4^3 + 5^3 = 6^3$.

This dissection is specific to the solution $(a,b,c;d)=(3,4,5;6)$ of the Diophantine equation $a^3+b^3+c^3=d^3$; I don't know whether an $8$-piece dissection is possible for any other solution. In some cases it's clearly not possible: if $a<b<c$ and $a+c<d$ then there's at least one corner of the $d$-cube, say $(1,1,1)$, that contributes to the $a$-cube, but then any cell  $(x,y,z)$ with $\max(x,y,z) = a+1$ cannot connect to any corner. This first happens for $(a,b,c;d) = (6,32,33;41)$.

What's the smallest dissection for the "taxicab" identity $1^3 + 12^3 = 9^3 + 10^3$?

JHI's elegant lower bound of $8$ on $N$ is achieved by an explicit dissection. I show my construction below; you might want to try to find a solution yourself before proceeding $-$ it makes for a neat puzzle. There may well be other ways to do it.

If somebody can make a "$3$-dimensional" graphic or picture of the $8$-piece dissection, you're welcome to add it by editing my answer. My diagrams are two-dimensional, labeling each piece with its height. Fortunately the dissection is simple enough for this to be possible; in particular, the eight pieces comprise four boxes and four L-shaped prisms. This also made it possible to find the solution using just pencil and paper on an otherwise uneventful international flight.

Begin by cutting the $6 \times 6 \times 6$ cube top to bottom into three pieces, as shown in top view in the first square diagram. Then cut each piece horizontally in two, dividing AB into $3+3$,$\phantom.$ C into $4+2$, and D into $5+1$. Each AB piece is then further subdivided into a box B and an L-shaped prism A. The second diagram shows (say) the bottom layer of four pieces, and the third diagram shows the top. Note that the AB subdivisions are not quite the same.

Pieces with the same color will come together to form a smaller cube. The larger C piece is a $4$-cube, and the two A pieces form a $3$-cube as shown. It remains to construct the $5$-cube from the remaining five pieces. The last two diagrams show the bottom and top of the $5$-cube.

The two $5$'s are the larger B piece, rotated to span the entire height of the cube, and the thick D piece. The thin D piece completes the bottom, with width $1$. The top is filled by the thinner C piece and the smaller B, both rotated to height 4. QEF

I guess that a physical model won't make for a hard puzzle to reconstitute into either one or three cubes (e.g. the AB, C, and D parts of the $6$-cube are independent) but would still make a nice physical model of the identity $3^3 + 4^3 + 5^3 = 6^3$.

This dissection is specific to the solution $(a,b,c;d)=(3,4,5;6)$ of the Diophantine equation $a^3+b^3+c^3=d^3$; I don't know whether an $8$-piece dissection is possible for any other solution. JHI's analysis shows that one can never get below $8$, and in some cases even that's not possible: if $a<b<c$ and $a+c<d$ then there's at least one corner of the $d$-cube, say $(1,1,1)$, that contributes to the $a$-cube, but then any cell  $(x,y,z)$ with $\max(x,y,z) = a+1$ cannot connect to any corner. This first happens for $(a,b,c;d) = (6,32,33;41)$.

What's the minimal dissection for the "taxicab" identity $1^3 + 12^3 = 9^3 + 10^3$? JHI's corner-cutting argument shows that at least nine pieces are needed.