Advantage of metropolis hastings or MonteCarlo methods over a simple grid search?
When the search space becomes larger, it can become infeasible to do an exhaustive search. So we turn to Monte Carlo methods out of necessity.
I have a relatively simple function with three unknown input parameters for which I only know the upper and lower bounds. I also know what the output Y should be for all of my data. 
So far I have done a simple grid search in python, looping through all of the possible parameter combinations and returning those results where the error between Y predicted and Y observed is within a set limit.  
I then look at the results to see which set of parameters performs best for each group of samples, look at the trade-off between parameters, see how outliers effect the data etc.. 
So really my questions is - whilst the grid search method I'm using is a bit cumbersome, what advantages would there be in using Monte Carlo methods such as metropolis hastings instead?
I am currently researching into MCMC methods, but don’t have any practical experience in using them and, in this instance, can’t quite see what might be gained.
I’d greatly appreciate any comments or suggestions
Many Thanks