We prove continuity results for Fourier integral operators with symbols in modulation spaces, acting between modulation spaces. The phase functions belong to a class of non-degenerate generalized quadratic forms that includes Schrödinger propagators and pseudodifferential operators. As a byproduct, we obtain a characterization of all exponents p, q, r₁, r₂, t₁, t₂∈[1, ∞] of modulation spaces such that a symbol in M^{p, q}(ℝ^2d) gives a pseudodifferential operator that is continuous from M^r₁,r₂(ℝ^d) into M^t₁,t₂(ℝ^d).