How about sensibility, no hint of sex here.

sensibility (the ability to perceive or feel) or sensuality (quality or state of being sensual), I would prob. choose the latter, but it depends on the centext, I think
-- as tony said, sensibility is "safer"

Somebody wrote, “Sensibility (the ability to perceive or feel) or sensuality (quality or state of being sensual), I would prob. choose the latter, but it depends on the context, I think
-- as tony said, sensibility is "safer."

I don’t go along with this suggestion. Jane Austen wasn't referring to our five senses but instead to emotion i.e. clearly a word not related to the five senses but to feelings.

According to Wikipedia:
"Elinor and Marianne Dashwood are sisters with opposite temperaments. Traditionally, it has been viewed that 19 year old Elinor, the elder daughter, represents "sense" (reason) of the title, and Marianne , who is 17, represents "sensibility" (emotion). However this view is a very restricting one. On close inspection of the novel it can be seen that each sister represents different aspects of each characteristic."
(http://en.wikipedia.org/wiki/Sense_and_Sensibility)

However, Erik, who started the discussion, wrote: "I need a translation for Sinnlichkeit that doesn't have a sexual connotation, but rather purely relates to the 5 senses. "

I need such a word myself in a translation to describe an ambient music group, OTR, whose music may heighten people’s perceptions through the five senses. Given that there is no single word I shall fall back on Wolf Friederich’s helpful concept of Hendiadyoin (“Handy to hand on” TG) or “eins durch zwei”*; i.e. where no one single word is adequate one takes two words or even more.

So here
“sensory awareness / consciousness / perception or even alertness” -- you take your pick -- seem to be better than for instance “sensory or sensorial nature” as in ‘Collins German Dictionary’ (London 1991).
Typically I completely underestimated the widespread use of “sensorial”; Google gives 844,000 hits for it in English, and over 2.5 million in Spanish and Portugese combined.

*p.46 Technik des Übersetzens’ (Hueber 1977 u. 1995 Munich)