Singular or plural verb after a series connected by "or"

As explained in this answer, the accepted rule is that the verb agrees in number with whichever disjoint subject lies closest to it. For example, from the Purdue Online Writing Lab:

When a compound subject contains both a singular and a plural noun or pronoun joined by or or nor, the verb should agree with the part of the subject that is nearer the verb.

So in your case, following that rule would have you writing it this way:

A can of sardines, radio batteries, or a bicycle is more precious than money.

That actually sounds right to my ear. Sure, you may not be able to satisfy all the people all the time, but then again, who ever can? :)

It becomes more obvious which choice sounds best if phrased as a question with a slight change:

Which one of these three choices is more precious than money: a can of sardines, some radio batteries, or a bicycle?

It becomes especially difficult to find something to suit all tastes when the closest subject is I and the verb is be. For example, with modified examples from this question:

1. Neither you nor I am to blame.
2. Neither you nor I are to blame.
3. Neither you nor I is to blame.
4. Neither one of you or me is to blame.

Choosing 2 there is common enough, but a copyeditor will "fix" it for you in a trice. The formal rule would have you elect choice 1 above, but it can make some people uncomfortable who are unfamiliar or simply disagree with the formal rule. You may not think 3 would be something people would ever choose, but a slight modification leading to 4 is easily enough done.

There's a lesson in that. If something makes people uncomfortable to say, hear, or read, it may be worth taking the trouble to reword it to avoid discomforting them.

It's the mixture of singulars and plurals that's making the construction awkward. How about this instead?

Cans of sardines, radio batteries, and bicycles are more precious than money.

Even this leaves potential for confusion (it's not clear if "cans" applies to sardines only or to radio batteries and even bicycles), so this would be clearer again:

Radio batteries, cans of sardines, and bicylces are more precious than money.

The serial comma is a matter of taste :-)

I'd have to agree with the commenter above (@JohnLawler); there's no rule that will satisfy everyone. On the other hand you could rewrite your sentence slightly to make your point. In the example below I've changed 'radio batteries' to a singular item as it fits the series better (and avoids faulty parallelism). I've also inserted a phrase:

A can of sardines, a radio battery, a bicycle; any one of these is more precious than money.

NOTE: Some of the comments below refer to an earlier version of this answer. They might still prove instructive in answering the original question so I've opted not to remove them for now.