A broadly, continuously and rapidly tunable cw laser spectrometer based on difference-frequency generation in bulk LiNbO3 is reported. Using an external cavity diode laser, tunable from 795 to 825 nm, and a cw diode-pumped Nd:YAG ring laser at 1064 nm, we produced narrowband mid-ir laser light between 3.16 and 3.67 mu m. This broad tuning range of 440 cm(-1) can be critically phase-matched by changing the external crystal angle within a range of only 0.4 degrees. Even for a fixed crystal angle broadband phase-matching over 230 cm(-1) was achieved. No realignment was required when tuning the laser over the whole wavelength range, which enabled the use of a compact 36-m multipass cell as gas chamber. The conversion efficiency could be improved by almost a factor of 3 by applying noncollinear instead of collinear phase-matching, resulting in 30 nW of idler power with good beam quality. Spectra of methane were recorded in laboratory air, which demonstrate the rapid and continuous broad tunability at high sensitivity, enabling sub-ppm detection. Finally, wavelength modulation spectroscopy at high resolution was applied as a promising tool for further inproving the performance of this laser spectrometer.