Aims: We investigate the photometric modulation induced by magnetic activity cycles and study the relationship between rotation period and activity cycle(s) in late-type (FGKM) stars. Methods: We analysed light curves, spanning up to nine years, of 125 nearby stars provided by the All Sky Automated Survey (ASAS). The sample is mainly composed of low-activity, main-sequence late-A to mid-M-type stars. We performed a search for short (days) and long-term (years) periodic variations in the photometry. We modelled the light curves with combinations of sinusoids to measure the properties of these periodic signals. To provide a better statistical interpretation of our results, we complement our new results with results from previous similar works. Results: We have been able to measure long-term photometric cycles of 47 stars, out of which 39 have been derived with false alarm probabilities (FAP) of less than 0.1 per cent. Rotational modulation was also detected and rotational periods were measured in 36 stars. For 28 stars we have simultaneous measurements of activity cycles and rotational periods, 17 of which are M-type stars. We measured both photometric amplitudes and periods from sinusoidal fits. The measured cycle periods range from 2 to 14 yr with photometric amplitudes in the range of 5-20 mmag. We found that the distribution of cycle lengths for the different spectral types is similar, as the mean cycle is 9.5 yr for F-type stars, 6.7 yr for G-type stars, 8.5 yr for K-type stars, 6.0 yr for early M-type stars, and 7.1 yr for mid-M-type stars. On the other hand, the distribution of rotation periods is completely different, trending to longer periods for later type stars, from a mean rotation of 8.6 days for F-type stars to 85.4 days in mid-M-type stars. The amplitudes induced by magnetic cycles and rotation show a clear correlation. A trend of photometric amplitudes with rotation period is also outlined in the data. The amplitudes of the photometric variability induced by activity cycles of main-sequence GK stars are lower than those of early- and mid-M dwarfs for a given activity index. Using spectroscopic data, we also provide an update in the empirical relationship between the level of chromospheric activity as given by log10R 'HK and the rotation periods.