Linear receivers are often used to reduce the implementation complexity of multiple-antenna systems. In a traditional linear receiver architecture, the receive antennas are used to separate out the codewords sent by each transmit antenna, which can then be decoded individually. Although easy to implement, this approach can be highly suboptimal when the channel matrix is near singular. This paper develops a new linear receiver architecture that uses the receive antennas to create an effective channel matrix with integer-valued entries. Rather than attempting to recover transmitted codewords directly, the decoder recovers integer combinations of the code- words according to the entries of the effective channel matrix. The codewords are all generated using the same linear code, which guarantees that these integer combinations are themselves codewords. Provided that the effective channel is full rank, these integer combinations can then be digitally solved for the original codewords. This paper focuses on the special case where there is no coding across transmit antennas and no channel state information at the transmitter(s), which corresponds either to a multiuser uplink scenario or to single-user V-BLAST encoding. In this setting, the proposed integer-forcing linear receiver significantly outperforms conventional linear architectures such as the zero forcing and linear minimum mean-squared error receiver. In the high signal-to-noise ratio regime, the proposed receiver attains the optimal diversity-multiplexing tradeoff for the standard multiple-input multiple-output (MIMO) channel with no coding across transmit antennas. It is further shown that in an extended MIMO model with interference, the integer- forcing linear receiver achieves the optimal generalized degrees of freedom.