Compressive random projections followed by l₁ reconstruction is by now a well-known approach to capturing sparsely distributed information, but applying this approach via discretization to estimation of continuous-valued parameters can perform poorly due to basis mismatch. However, we show in this paper it is still possible to capture the information required for effective estimation using a small number of random projections. We characterize the isometries required for preserving the geometric structure of estimation in additive white Gaussian noise (AWGN) under such compressive measurements. Under these conditions, estimation-theoretic quantities such as the Cramer- Rao Lower Bound (CRLB) are preserved, except for attenuation of the Signal-to-Noise Ratio (SNR) by the dimensionality reduction factor. For the canonical problem of frequency estimation of a single sinusoid based on N uniformly spaced samples, we show that the required isometries hold for M = O(log N) random projections, and that the CRLB scales as predicted. While we prove isometry results for a single sinusoid, we present an algorithm to estimate multiple sinusoids from compressive measurements. Our algorithm combines coarse estimation on a grid with iterative Newton updates and avoids the error floors incurred by prior algorithms which apply standard compressed sensing with an oversampled grid. Numerical results are provided for spatial frequency (equivalently, angle of arrival) estimation for large (32 × 32) two-dimensional arrays.