The sixteenth blessing of the Shmoneh Esrei says, שמע קולנו...וקבל ברחמים וברצון את תפילתנו: "Hear our voice...and accept our prayer with compassion and favor."
It seems redundant and circular to ask Hashem to answer our prayers. If he was not going to answer our prayers, why would he answer this one?
What does this blessing accomplish?
Here's one possible answer, from the commentary Maggid Tzedek on the siddur (link to page):
כדאיתא במסכת תענית (דף כה:): שמואל הקטן גזר תעניתא וירדו להם גשמים קודם הנץ החמה. כסבורין העם לומר שבחו של צבור הוא. אמר להם: אמשול לכם משל למה הדבר דומה, לעבד שמבקש פרס מרבו, אמר להם: תנו לא ואל אשמע קולו
As is related in Masekhet Ta'anit (page 25b): Shmuel HaKatan decreed a fast day [because of drought] and rain fell [in the early morning of the decreed fast day] before sunrise [and therefore before the public had even started offering their fast day prayers]. The people thought that this spoke to the merit of the community [i.e., the granting of their desire so quickly]. He [Shmuel HaKatan] said to them, I'll offer a parable regarding what has happened: a servant sought a boon from his master. He [the master] said: Give him what he wants but just don't make me have to hear his voice.
So too here, we express the desire that, outside of any actual granting of our requests, the voice of our asking itself be privileged to be desired and not obnoxious.... The commentator goes on to offer several other answers, all different and all interesting.
Assuming we are referring to the requests we've made previously, there are still ways in which a request can be granted. Indeed we do not implore God "accept our prayers", but "pity us and accept our prayers mercifully and amiably". We could at very least be asking that our requests be granted in a nice way rather than a scornful one.
This question is based on the assumption that the reason a prayer wouldn't be accepted is because it isn't heard. However, Hashem hears it all. While you may not be Zoche for that request to be answered, you might be Zoche for this request to be answered. The Gemara in Sanhedrin 44b specifically says that a person should always Daven that there be no obstacles in acceptance of his Teffila. The Maharal in Nesiv Hoavodah 12 explains this a bit.
The ערבי נחל (I think) and other Chassidishe Sefarim explain that when we Daven we are clearing a clogged pipe, so to speak, because of some Kitrug. When we pray, we are opening up these channels, by means of our realization of how we are completely dependent on Hashem. See מהר"ל נתיב העבודה
Therefore, even if a Tefilla won't do the job in its specific area, we ask a separate request to accept our Tefilla. If this one does its job it will help for the other in retrospect. Think of how you can run a debugging program that will fix the computer it is running on.
We find this in the Pasuk in Shemos where it says Hashem heard their cries, remembered the Avos, saw their plight and knew. Later there is a different order because of the context. But we see that there are certain Zechusim needed for the acceptance of a prayer. We can pray for that, too.
I don't know the source, but I remember hearing a medrash that states that when a tefillah is not answered affirmatively, the tefillah is still "stored up" as a merit for the person or klal Yisrael. It's possible that we are not asking for Hashem to respond to our prayers, as we do not say "answer" or "respond" but just that they be accepted.
