There's an obstruction to embedding $n$-complexes in $\mathbb R^{2n}$ provided $n \geq 3$ due to Shapiro.
MR0089410 (19,671a) Shapiro, Arnold Obstructions to the imbedding of a complex in a euclidean space. I. The first obstruction. Ann. of Math. (2) 66 (1957), 256–269.
or as Ian suggests, van Kampen pre-dates Shapiro.
JFM 58.0615.03 van Kampen, E. R. Berichtung zu:``Komplexe in euklidischen Räumen''. (German) Abhandlungen Hamburg 9, 152-153 (1932).
Could you be more precise on how you want to modify the your complexes that originally are in $\mathbb R^3$, your special case?