I'd like to answer this with two separate examples related to the current context of human emotions.
I recently read a mostly entertaining (and occasionally eye-opening) little book by the well-known animal behaviorist Mark Bekoff called The Emotional Lives of Animals. Bekoff stands in stark contrast to objective, evidence-based scientific research on animals and a conservative, hard-nosed behaviorist community ( a la Skinner & friends) by advocating that non-human animals - ranging from self-aware primates to corvids and parrots down to rodents and possibly reptiles - are not simply mechanical, sensory-driven organisms, but richly diverse in feeling and identity. His storytelling is heavily based on anecdotes and personal experiences, which he argues tells a fairer story of emotive animal life, much like psychology for humans explores the paramechanical minds of humans. One story focused on elephant matriarchy, caring for calves, and the seemingly culture-like community that includes mourning and funerals to graveyards and holding/sharing multi-generational memories of helpful humans. In this section he claimed that elephants are notorious for feeling (with a capital F) extraordinarily loudly and deeply and that researchers have categorized types of feeling wholly alien in their complexity compared to humans.
My second example is a much more serious exploration of cognition and feeling: Peter Godfry-Smith's Other Mind: The Octopus, the Sea, and The Deep Origins of Consciousness. Again focusing on feeling animals, Godfry-Smith sheds light on mollusc evolution and the currently enigmatic minds of octopus, squid, and the startling cuttlefish. The author is time and again enamored by first-hand accounts of diving in Octopolus and being led tentacle in hand (literally) on a tour; unusual battles between territorial males; and most impressively, the convergent evolution of complex emotion, what he terms fuzzy, mental "static," and the vivid and colorful signaling of what he turns to philosophy to describe as an expression of the alien mind without. His use of anecdotes is similarly consistent, but he peppers his account throughout with related studies and philosophy. The most memorable section of the book deals with his relationship with cuttlefish and how he resists in all but words to define their awareness and how they respond to another feeling, thinking mind. The brief stories he tells portray these - what at one point when illustrating octopus motility and the disconnected nervous system, as brains floating in the sea - cuttlefish as feeling internally, expressing in bright colors externally (consciously or not?), and when they decide to suddenly recognize or completely ignore your existence altogether.
What Bekoff and Godfry-Smith illustrate in their vaguely disparate accounts is that pan-species emotion exists and is not only comparable to humans, but sometimes like in the case of Orca and elephants, may exceed it in neural complexity. Bekoff suggests and believes fairly strongly that recognizable emotions in animals are not anthropomorphisms (in fact, he encourages people to identify and sometimes label shared feelings with pets and animals), but divergent forms of similar emotional expression. Godfry-Smith, on the other hand, is amazed at how something so specific i.e. like maternal nurturance or communitarian affection to mammals can arise in molluscs, who are alien minds by contrast, and evolved through hundreds of millions of years as separate individuals, without community. My own opinion about human emotions straddles this line: feeling is basic sensory response interpreted by self-awareness (as opposed to temporary or fleeting self-awareness in molluscs). To complicate it further, humans are highly social and cooperative struggling with a mind-body-language divide and identify (or need) to exist as individual and member of a community, never separate. So feeling is both internal and sympathetic, like minds reaching across the opaque deep.
I definitely value reason/logic as well, but I think that emotions have a very important and necessary role to play in our lives. Valuing emotion gets a bad rap because people associate emotion with lack of control, impulsive and reckless. Something to ignore in favor of a level head. And sometimes emotion is like that, not even the most logical person can avoid that feeling, though they'd try. But overall, if you regulate and manage your emotions they wont control your life in a negative fashion, they'll enhance it and balance it. We need to allow ourselves to feel, to go through things, in turn that vulnerability can make us stronger and wiser. So we can understand how to deal with things instead of shutting off. 
