Submillimetre galaxies (SMGs) are some of the most extreme star-forming systems in the Universe, whose place in the framework of galaxy evolution is as yet uncertain. It has been hypothesized that SMGs are progenitors of local early-type galaxies, requiring that SMGs generally reside in galaxy cluster progenitors at high redshift. We test this hypothesis and explore SMG environments using a narrow-band VLT/HAWK-I+GRAAL study of H $\alpha$ and [O III] emitters around an unbiased sample of three ALMA-identified and spectroscopically confirmed SMGs at $z \sim 2.3$ and $\sim 3.3$, where these SMGs were selected solely on spectroscopic redshift. Comparing with blank-field observations at similar epochs, we find that one of the three SMGs lies in an overdensity of emission-line sources on the $\sim 4$ Mpc scale of the HAWK-I field of view, with overdensity parameter $\delta _{g} = 2.6^{+1.4}_{-1.2}$. A second SMG is significantly overdense only on $\lesssim 1.6$ Mpc scales and the final SMG is consistent with residing in a blank field environment. The total masses of the two overdensities are estimated to be $\log (M_{h}/{\rm M}_{\odot }) =$ 12.1-14.4, leading to present-day masses of $\log (M_{h,z=0}/{\rm M}_{\odot }) =$ 12.9-15.9. These results imply that SMGs occupy a range of environments, from overdense protoclusters or protogroups to the blank field, suggesting that while some SMGs are strong candidates for the progenitors of massive elliptical galaxies in clusters, this may not be their only possible evolutionary pathway.