Coupled one-dimensional photochemical-climate calculations have been performed for hypothetical Earth-like planets around M dwarfs. Visible/near-infrared and thermal-infrared synthetic spectra of these planets were generated to determine which biosignature gases might be observed by a future, space-based telescope. Our star sample included two observed active M dwarfs-AD Leo and GJ 643-and three quiescent model stars. The spectral distribution of these stars in the ultraviolet generates a different photochemistry on these planets. As a result, the biogenic gases CH4, N2O, and CH3Cl have substantially longer lifetimes and higher mixing ratios than on Earth, making them potentially observable by space-based telescopes. On the active M-star planets, an ozone layer similar to Earth's was developed that resulted in a spectroscopic signature comparable to the terrestrial one. The simultaneous detection of O2 (or O3) and a reduced gas in a planet's atmosphere has been suggested as strong evidence for life. Planets circling M stars may be good locations to search for such evidence.