I understand that one's accent will vary depending on one's nationality, but I have heard some people speak without the trilled R and have wondered about that letter in particular. I personally roll the R, but I also understand that there may be some people who simply cannot produce that sound. What is the officially recognized pronunciation of the letter R? How strict was Zamenhof on its being trilled?
The most important grammatical references, PMEG and PAG (Plena Analiza Gramatiko), both recommend R to be pronounced as an alveolar trill [r] (i.e. trilled with the tip of the tongue). PMEG says that a uvular trill [R] (like the French R) is also acceptable.
Apperently Zamenhof himself was not very strict about this. In the five version of the grammar rules of the Fundamento de Esperanto written in French, English, German, Russian and Polish, he always just explained the pronunciation of R by equating it with the pronunciation of the R in the language, in which he was explaining it. This could justify quite a variety of different ways to pronounce the R.
But already very early the opinion, that the alveolar trill is the best pronunciation, started becoming popular, and this opinion has become part of the Esperanto pronunciation standard. Here is what the Esperanto Wikipedia writes about this:
La Fundamento de Esperanto samtempe egaligis la Esperantan /r/ kun la sufiĉe diversaj /r/-oj de la lingvoj angla, franca, germana, pola kaj rusa, kaj sekve lasis pli da libereco por la prononco de /r/; sed jam baldaŭ aperis la opinio ke nur la vibra langopinta [r] (kiel en la rusa kaj pola, kaj aldone ankaŭ en la itala) estas bona prononco, kaj tiu opinio iĝis parto de la nun firmiĝinta prononca normo.
So if one can pronounce the alveolar trill, one should certainly use it in Esperanto. If one can't pronounce it, but one can pronounce the uvular trill, one can use that as an acceptable alternative. If one can't pronounce either trill, one should make an effort to learn at least one of them (preferably the alveolar one) instead of just contenting oneself with whatever R one can easily pronounce.
In Fundamento de Esperanto is stated how the r should be pronounced in several languages. In the French and German ones it is simply stated that the r be pronounced as the regular r in those languages. This can be a variety of things, but usually is a uvular approximant or fricative [ʁ] or [ʁ̝], or a uvular trill [ʀ] (all voiced).
In the English version it reads that the r be pronounced as the ‘r’ in the word ‘rare’, which would be an alveolar approximant [ɹ].
Thus it seems that a wide variety of r-sounds is officially accepted.
The other answers already explained that you can pronounce r whichever way seems natural to you, and it's always correct. But that among the correct pronunciations, the alveolar trill (as in Italian, for example) is considered the best. I just want to add the historical background which explains why this is the most natural approach for an international language rooted in Europe.
Until the 17th century, the alveolar trill was the normal educated pronunciation of r almost everywhere in Europe, even including England and France. There seems to be a natural tendency for this sound to change to one that is easier to produce, and this has happened in different directions and at different speed in different places. Whereas some languages outside Europe have several variants at the same time and one needs to distinguish them, this is not the case anywhere in Europe. Therefore all ways to produce an r-like sound are considered equivalent - except for the fact that the one that requires the most effort and produces the clearest results, which is the original alveolar trill, is considered the best of the variants and is taught to singers and professional speakers in many regions where it has otherwise fallen out of use. And of course most languages have one or more ways they normally pronounce r.
E.g. in my native German, the r of most speakers is similar to French r, but some regions have different standards (r trilled or not, produced in various places), and the alveolar trill is taught for use in classical music and in theatre.