We present results from a new Keck spectroscopic survey of UV-faint LBGs in the redshift range 3<z<7. Combined with earlier Keck and published ESO VLT data, our sample contains more than 600 dropouts, offering new insight into the nature of sub-L* sources typical of those likely to dominate the cosmic reionisation process. Here we use this sample to characterise the fraction of strong Lya emitters within the continuum-selected dropouts. By quantifying how the "Lya fraction" varies with redshift, we seek to constrain changes in Lya transmission associated with reionisation. In order to distinguish the effects of reionisation from other factors which affect the Lya fraction (e.g. dust, ISM kinematics), we study the luminosity and redshift-dependence of the Lya fraction over 3<z<6, when the IGM is known to be ionised. These results reveal that low luminosity galaxies show strong Lya emission much more frequently than luminous systems, and that at fixed luminosity, the prevalence of strong Lya emission increases moderately with redshift over 3 < z < 6. Based on the correlation between blue UV slopes and strong Lya emitting galaxies in our dataset, we argue that the Lya fraction trends are governed by redshift and luminosity-dependent variations in the dust obscuration, with likely additional contributions from trends in the kinematics and covering fraction of neutral hydrogen. We find a tentative decrease in the Lya fraction at z~7 based on the limited IR spectra for candidate z~7 lensed LBGs, a result which, if confirmed with future surveys, would suggest an increase in the neutral fraction by this epoch. Given the supply of z and Y-drops now available from Hubble WFC3/IR surveys, we show it will soon be possible to significantly improve estimates of the Lya fraction using optical and near-IR spectrographs, thereby extending the study conducted in this paper to 7<z<8.