We consider the maximum (weight) independent set problem in unit disk graphs. The high complexity of the existing polynomial-time approximation schemes motivated the development of faster constant-approximation algorithms. In this article, we present a 2.16-approximation algorithm that runs in O(n log^2 n) time and a 2-approximation algorithm that runs in O(n^2 log n) time for the unweighted version of the problem. In the weighted version, the running times increase by an O(log n) factor. Our algorithms are based on a classic strip decomposition, but we improve over previous algorithms by efficiently using geometric data structures. We also propose a PTAS for the unweighted version.