All Rs are voiced. Is there any reason why they don't?
Writing words in katakana is guided by pronunciation, and accents can affect pronunciation, which is why there are often a couple choices, especially if the word isn't commonly used.
Just as in some English accents where the syllable-final R's get dropped (imagine how a British person might say the examples Thomas gave above), Japanese doesn't natively use syllable-final consonants except for the moraic nasal (often realized as an N)... Except for that mosaic nasal, you will never see two consonants right next to each other in Japanese (edit: OK I just thought of some phonetic exceptions like "tsu" or devoiced high vowels, but that's a whole other discussion)
Some examples:
The word パンク (panku), which could refer to punk music or a flat tire (short for puncture, I suppose?), shows the moraic nasal next to the [k], but whether it's voiced or not, there must be a vowel after the k... This is because there's no phonetic character in Japanese for "k" by itself, only "ka," "ki," "ku," "ke," and "ko."
With the R, the brand Burberry is written バーバリー (baabarii) where the long vowel is used instead of inserting a ル (バルバリー, barubarii), which would sound less like the word they're trying to approximate, and also throw off the cadence of the word.
With the word "culture," you end up with both treatments for syllable-final consontants, and we get カルチャー, karuchaa.
Sometimes loan words are affected by the accents of the languages they're borrowed from: you can write the anglicized parmesan as パルメザン (parumezan), but the Italian parmigiano is written パルメジャーノ (parumejaano). Perhaps because the source language is Italian, both variations use the "ru" instead of a long vowel for the R, but other sounds get changed to approximate the Italian accent more closely.
That's all to just agree that... it just depends on how it's pronounced