Due to the computational intractibility of many mixed-integer PDE-constrained optimization problems, there is still significant potential for improvement in solver performance through the use of heuristics. Starting from the successful problem-unspecific "sum-up rounding" scheme for ODE-constrained optimal control problems developed by Sager et al. between 2006 and 2012, we put forward a family of similar nearest-neighbor rounding schemes for PDE-constrained optimization problems. We show that detailed geometric analysis can yield problem-specific accuracy guarantees similar to those given by sum-up rounding. Finally, by applying our nearest-neighbor rounding schemes to a set of test problems from the domains of distributed control, boundary control and optimal design, we show that our approach produces viable results and may warrant further investigation.