Jewish holidays are generally distinguished by their focus on the Temple. Pesach, Sukkot, and Shavuot are occasions of obligatory pilgrimage and sacrifice, while Yom Kippur focuses on the day's special Temple service. Each holiday has special communal sacrifices, and on these days we are commanded to blow trumpets in the Temple. The holiday prayer "yaale veyavo" is recited only in those blessings - Retzeh (prayer) and Boneh Yerushalayim (birkat hamazon) - whose subject is the Temple. And so on and so on.
What, then, are we to do these days, when the Temple is destroyed? Instead of being occasions of celebration, should not holidays become days of pain, when we are directly confronted with what we have lost? Tisha Beav - now a day of mourning - will turn into a festive day once the Temple is rebuilt; wouldn't it be appropriate for holidays to be commemorated the same way?
My answer to this is no, and the explanation comes in two parts. First, to explain what the purpose of holidays is, beyond the particular rituals we can no longer perform. And second, to explain how that purpose is still fulfilled today.
First: Holidays were an opportunity for closeness with God. Of course, God is "found" in every place, and there are many opportunities for closeness unrelated to Temple. But the atmosphere of holidays, the prohibition on work, the physical closeness to "God's house", and the special rituals and celebrations, made holidays especially good opportunities for individuals to approach God. Moreover, as the entire nation celebrated together, holidays became occasions of national closeness to God.
Second: Despite the loss of the Temple, I believe we can still achieve this special holiday closeness. In the Temple, Divine closeness was achieved through certain rituals, like the Yom Kippur sacrifice with precipitated the forgiveness of our sins. Those rituals were a commandment, and the resulting closeness was the reward. We (the religious community) still want to perform these rituals, but we are prevented by circumstances from doing so. Halachically, this situation is defined as a situation of "oness". In such a situation, having done everything we are capable of doing, we are still entitled to the reward.
One may argue that this is not a situation of complete "oness", since the Temple's destruction is caused by our sins, and by repenting we could cause it to be rebuilt. This is true on one level, but false on another. The process of returning to Israel, appointing a king, rebuilding the Temple, and offering sacrifices takes a certain amount of time. We mourn all year for the Temple, because we have this time yet are not using it appropriately in order to speed up redemption. But once the holiday has begun, it is no longer possible to do all the tasks needed to rebuild the Temple. Regarding that particular holiday, we are truly in a state of "oness". We still perform whatever rituals we are capable of, and God still rewards us with His closeness as if we had performed all the other rituals as well.