The InvenSense MPU-6050 sensor contains a MEMS accelerometer and a MEMS gyro in a single chip. It is very accurate, as it contains 16-bits analog to digital conversion hardware for each channel. Therefor it captures the x, y, and z channel at the same time.
The MPU-6050 is not expensive, especially given the fact that it combines both an accelerometer and a gyro.
1 x MPU-6050 3-Axis Accelerometer Module