Which varieties are sums of tensor powers of the Lefschetz motive? Any smooth projective variety $X$ gives an object $h(X)$ in the category of pure Chow motives.   If $X$ is a generalized flag variety, i.e. a quotient $G/P$ where $G$ is semisimple linear algebraic group and $P$ is a parabolic subgroup, I believe $h(X)$ is a direct sum of tensor powers of the Lefschetz motive, because $X$ can be decomposed into Schubert varieties which are copies of $\mathbb{A}^n$ for various $n$.
If this is correct, I'd like to know: which other smooth projective varieties give pure Chow motives that are direct sums of tensor powers of the Lefschetz motive?
My intuition is that any variety with something like a "Schubert decomposition" — roughly, a well-behaved way of expressing it as a disjoint union of copies of $\mathbb{A}^n$'s — will have this property.   I feel there should be plenty.  But I don't actually know any varieties with this property, apart from flag varieties!
Any variety $X$ of dimension $d$ over $\mathbb{F}_p$ having this property will have an associated polynomial $N_X$ of degree $d$ with natural number coefficients:
$$ N_X(q) = \sum_{n = 0}^d a_n q^n $$
such that $X$ has $N_X(q)$ points over $\mathbb{F}_q$ when $q$ is any power of $p$.
Is the converse true?  Is a smooth projective variety $X$ over $\mathbb{F}_p$ with a polynomial $N_X$ having this property always a direct sum of tensor powers of the Lefschetz motive?
 A: One class of examples is already indicated in the comments, and the question itself. I thought it would be good to include this in an official answer.

Proposition. Let $X$ be smooth projective variety over a field $k$, such that there exists a chain of closed sets
$$X=X_n\supset X_{n-1}\supset \ldots X_{-1}=\emptyset$$
such that $X_i-X_{i-1}$ is a union of affine spaces or split tori (products of $\mathbb{G}_m$'s). Then the Chow motive is of Tate type, i.e. sum of powers of the Lefschetz motive.


Cor. The conclusion holds for flag varieties, and projective toric varieties.

The proposition follows from the  distinguished triangles
$$ M_{gm}^c(X_{i-1})\to M_{gm}^c(X_{i})\to M_{gm}^c(X_{i}-X_{i-1})$$
where $M_{gm}^c$ is the motive with compact support
in Voevodsky's category (cf. Voevodsky Triangulated categories of motives over a field and Mazza, Voevodsky, Weibel Lectures on motivic cohomology), induction, and the identification of $M_{gm}^c(X)$ with the Chow motive  of $X$ (up to translation and twist).
If you allow quasi projective varieties, then complements of hyperplane arrangements are also mixed Tate.
A few additional comments. As Ben Wieland and Will Sawin have pointed out, you get more examples by blowing an example you have of Tate type along  a subvariety with the same property. I can think of a few more examples off the top of my head, such that the Hilbert scheme of points on a rational surface.  On the other hand, I hope it's clear that examples would have to be quite special.
Over $\mathbb{C}$, the Hodge numbers $h^{pq}$ would vanish for $p\not=q$, so
most varieties would not satisfy this condition.
