How should I approach a situation when I try to apply some ML algorithms (classification, to be more specific, SVM in particular) over some high dimensional input, and the results I get are not quite satisfactory?

1, 2 or 3-dimensional data can be visualized, along with the algorithm's results, so you can get the hang of what's going on, and have some idea how to approach the problem. Once the data is over 3 dimensions, other than intuitively playing around with the parameters I am not really sure how to attack it?