Yes. The standard methods used to cross-validate and assess Kriging also apply, practically with no change, to almost any other method of interpolation. These include jackknifing, a form of leave-out-one cross-validation in which each data point is systematically removed from the dataset and predicted using the chosen variogram model. The discrepancy between the prediction, compared to the standard error of prediction, is a measure of (relative) accuracy of the predictor.
Obviously such cross-validation methods can be applied only to interpolators that supply some kind of quantitative measure of prediction error at each point. To apply them to other interpolators, you can still proceed as with kriging to estimate a variogram from the data. Use this variogram only to compute prediction standard errors, but use your chosen interpolator to do the predictions. Proceed as before. This ad hoc procedure is one reasonable and simple way to get a sense of how any interpolation procedure is performing.
You can also apply bootstrapping methods. They need to be specially adapted to spatial data to account for the expect spatial autocorrelation (so, once again--even if you are not using a kriging interpolator, you need to estimate a variogram). This is called spatial bootstrapping. It relies on repeated geostatistical simulation of an entire spatial dataset, using each replicate as a synthetic set of data with which to evaluate the accuracy of your chosen interpolator's predictions. After a large number of replications (usually at least 100 are chosen, more often 500 to 1000) the collected results of the prediction results can be assembled and assessed.
Notice that little can be done in the absence of some kind of quantitative model of the spatial autocorrelation (which is what the variogram provides), regardless of whether your chosen interpolator uses such a model.
This used to be a lot of work. (The first time I contemplated doing something like this, a quarter century ago, it took me several weeks to find and understand algorithms and write (Fortran) code to generate grids of simulated spatial data. Each simulation iteration would take minutes at a time on an 8 MHz PC souped up with a $1000 floating point coprocessor.) Today you can obtain, for free, with a quick Web search, tested code that performs such simulations almost instantaneously and use it to estimate variograms and collect thousands of simulation results within minutes.