Actually, this is a subtle and interesting question, and it is to some extent model dependent.
When one defines the integer graded homotopy groups, one does so using colimits over representation
spheres, so that ``RO(G)-graded information'' is encoded in the definition of homotopy groups. This
is why one can get away with defining weak equivalence in terms of the Z-graded homotopy groups
of all fixed point spectra.
As Justin says, $RO(G)$-graded homotopy groups are of definite interest. Calculating the integer
part of $RO(G)$ graded cohomology theories obscures structure that is only visible in the
$RO(G)$ graded world. But then the coefficients of the theory are the $RO(G)$-graded homotopy
groups of the representing spectrum.
The cells that build up $G$-spectra in the model theoretic sense are model dependent. Using
symmetric or orthogonal $G$-spectra, one must use cells of the form $G\times_H D^V$, and one
can restrict to G-representations. In Lewis-May or EKMM $G$-spectra, where all objects are
fibrant, one can use just cells $G/H_+\wedge D^n$. Here again, the $RO(G)$-graded information
is built into these $G$-spectra, which is why integer grading suffices.
In Tom Goodwillie's space level answer the notion of ``get'' needs interpretation. One can get every weak homotopy type of $G$-space using $G$-CW complexes defined using only cells of the form $G/H \times D^n$, no representations in sight. Intuitively, as Justin says, this works because of the classical theorem that representation spheres are themselves finite $G$-CW complexes in this sense, but that is irrelevant to the proof. If one wants only finitely many cells to get a smooth $G$-manifold, then one must use representations of all subgroups, as Tom says (and even then I don't know a good reference). EDIT added: This last sentence is nonsense. I'll quote from a Math Review:
The equivariant triangulation theorem for actions of compact Lie groups.
Math. Ann. 262 (1983), no. 4, 487–501.
``Let $G$ be a compact Lie group. The main results of the paper are the following.
Theorem 7.1: Let $M$ be a smooth $G$-manifold with or without boundary. Then there exists an equivariant triangulation of $M$. Corollary 7.2: Let $M$ be a smooth $G$-manifold with or without boundary. Then $M$ can be given an equivariant CW complex structure. The other results of the paper are the technical preparation for the above results. The paper is written very carefully and has a detailed history of results leading to the above ones.''
No representation spheres are needed. Where one needs representation spheres is
to construct dual triangulations of smooth compact manifolds and for work in
$RO(G)$-graded homology. See Chapter X, $G-CW(V)$ complexes and $RO(G)$-graded
cohomology, by Stefan Waner, in Equivariant homotopy and cohomology theory
(CBMS Conf. Series 91, AMS).
In any construction of the stable category of $G$-spectra, one wants to invert representation
spheres; for one of many reasons, as Justin says, Poincare' duality requires that. However,
when one inverts representation spheres, one actually inverts all homotopy spheres as well,
whether or not one wants to. See (Fausk, Lewis, May. The Picard group of the equivariant stable
homotopy category. Advances in Math 163(2001), 17--33). Philosophically, one might want to define homotopy groups using the entire Picard group of invertible objects, but that is calculationally unwise, to put it mildly. One can and should feel free to use either $Z$-graded or $RO(G)$-graded
homotopy groups, adapting the choice to the application at hand.