Have there been any scientific experiments attempting to communicate with one's unconscious mind? For the purposes of this question, I will define unconscious/subconscious as the parts of the human mind that are not normally available for introspection or control, but which are still involved in cognition. These parts of the mind might be involved in memory recall or assimilation of information. While I would define communication as receiving a recognizable response to input.

Have there been experiments attempting to establish communication with the researchers own unconscious mind? If so, what were their results? I'm not looking for "new age" kind of articles, but genuine scientific research findings.

The most direct way of communication that comes to mind would be talking to dream characters in a lucid dream and evaluating their responses upon awakening. Such characters are capable of semi-intelligent speech, and their responses at least casually appear to be related to the question at hand.  A consistent pattern of responses over several dreams might be considered a form of communication. 

Another experiment I can think of, which uses non-verbal communication would be using some daily activity to consistently triggers a recurring dream setting at night. For example, intense study causing dreams of classrooms or exams. In this case, the setting can be considered a response of unconscious system. 

Another example I can think of is exposing a person to a stimuli which consistently causes  some song to surface and "get stuck in one's head" - earworms during waking cognition. Alternatively, some saying or thought might arise unbidden in response to the stimuli. In this case, this can be considered another form of response. 

Have there been scientific experiments along the lines described above? What would be the term I would use to search for "communicating with unconscious" in a scientific setting?
In order to find such studies, it's a good idea to start off with mapping out what kind of situations would be relevant - that is, in what cases would it even be possible to communicate with someone's unconsciousnes?
Then you need to define what kind of communication you are interested in. 
Here are some ways to find some kind of unconscious communication: Sleepwalkers, hypnosis(highly controversial), vegetative states and the like, subliminal messages, facial expressions, etc. 
As such, the short answer to your question is "yes". The long answer is that your question is too wide.

Here's an article on using unconscious communication as a way of identifying narcissism:


  In this paper I suggest that analytical material implicitly carries in it a message proceeding from the unconscious and which is transmitted through representations. The message is expressed only when the object is recognized as such by the subject. In the narcissistic situation the unconscious message is not emitted. 
  http://europepmc.org/abstract/MED/3667086


An article on communicating with patients suffering from trauma:


  The interface among trauma, dissociation, and regression is discussed in the context of unconscious communication as a transferential enactment of unsymbolized experience.The view is offered that for certain patients in particular, past experience is not so much unconscious as “frozen in time” and that a key element of the psychoanalytic relationship is bridging dissociated aspects of self through the creation of a dyadic experiential field that is both “inside” and “outside.”
  http://www.tandfonline.com/doi/abs/10.1080/10481889109538911?journalCode=hpsd20#.VIAkl8mZGkU


Research on facial expressions:


  Studies reveal that when people are exposed to emotional facial expressions, they spontaneously react with distinct facial electromyographic (EMG) reactions in emotion-relevant facial muscles. These reactions reflect, in part, a tendency to mimic the facial stimuli. We investigated whether corresponding facial reactions can be elicited when people are unconsciously exposed to happy and angry facial expressions. Through use of the backward-masking technique, the subjects were prevented from consciously perceiving 30-ms exposures of happy, neutral, and angry target faces, which immediately were followed and masked by neutral faces. Despite the fact that exposure to happy and angry faces was unconscious, the subjects reacted with distinct facial muscle reactions that corresponded to the happy and angry stimulus faces. Our results show that both positive and negative emotional reactions can be unconsciously evoked, and particularly that important aspects of emotional face-to-face communication can occur on an unconscious level. 
  http://pss.sagepub.com/content/11/1/86.short