I once heard a joke (not a great one I'll admit...) about higher dimensional thinking that went as follows-

An engineer, a physicist, and a mathematician are discussing how to visualise four dimensions:
Engineer: I never really get it
Physicist: Oh it's really easy, just imagine three dimensional space over a time- that adds your fourth dimension.
Mathematician: No, it's way easier than that; just imagine $\mathbb{R}^n$ then set n equal to 4.

Now, if you've ever come across anything manifestly four dimensional (as opposed to 3+1 dimensional) like the linking of 2 spheres, it becomes fairly clear that what the physicist is saying doesn't cut the mustard- or, at least, needs some more elaboration as it stands.
The mathematician's answer is abstruse by the design of the joke but, modulo a few charts and bounding 3-folds, it certainly seems to be the dominant perspective- at least in published papers. The situation brings to mind the old Von Neumann quote about "...you never understand things. You just get used to them", and perhaps that really is the best you can do in this situation.
But one of the principal reasons for my interest in geometry is the additional intuition one gets from being in a space a little like one's own and it would be a shame to lose that so sharply, in the way that the engineer does, in going beyond 3 dimensions.
What I am looking for, from this uncountably wise and better experienced than I community of mathematicians, is a crutch- anything that makes it easier to see, for example, the linking of spheres- be that simple tricks, useful articles or esoteric (but, hopefully, ultimately useful) motivational diagrams: anything to help me be better than the engineer.
Community wiki rules apply- one idea per post etc.

I can't help you much with high-dimensional topology - it's not my field, and I've not picked up the various tricks topologists use to get a grip on the subject - but when dealing with the geometry of high-dimensional (or infinite-dimensional) vector spaces such as $\mathbb R^n$, there are plenty of ways to conceptualise these spaces that do not require visualising more than three dimensions directly.

For instance, one can view a high-dimensional vector space as a state space for a system with many degrees of freedom.  A megapixel image, for instance, is a point in a million-dimensional vector space; by varying the image, one can explore the space, and various subsets of this space correspond to various classes of images.

One can similarly interpret sound waves, a box of gases, an ecosystem, a voting population, a stream of digital data, trials of random variables, the results of a statistical survey, a probabilistic strategy in a two-player game, and many other concrete objects as states in a high-dimensional vector space, and various basic concepts such as convexity, distance, linearity, change of variables, orthogonality, or inner product can have very natural meanings in some of these models (though not in all).

It can take a bit of both theory and practice to merge one's intuition for these things with one's spatial intuition for vectors and vector spaces, but it can be done eventually (much as after one has enough exposure to measure theory, one can start merging one's intuition regarding cardinality, mass, length, volume, probability, cost, charge, and any number of other "real-life" measures).

For instance, the fact that most of the mass of a unit ball in high dimensions lurks near the boundary of the ball can be interpreted as a manifestation of the law of large numbers, using the interpretation of a high-dimensional vector space as the state space for a large number of trials of a random variable.

More generally, many facts about low-dimensional projections or slices of high-dimensional objects can be viewed from a probabilistic, statistical, or signal processing perspective.
