The Magid of Meziritch (Avodas Yisroel, Masei, Avos Chapter 2 &4) says that the period of "Bein Hamitzarim", or the 3 weeks, is one of the most opportune times to get close to Hashem. This is alluded to in a seemingly negative posuk right at the beginning of Eicha (1, 3):כָּל רֹדְפֶיהָ הִשִּׂיגוּהָ בֵּין הַמְּצָרִים – "all her pursuers overtook her within the straits". This can also teach the following: כָּל רדֶפ י ־ ה, "whoever chases after "Yud Kei" – Hashem", הִשִּׂיגוּהָ, "reaches Him", in the period of בֵּין הַמְּצָרִים.
This is analogous to a king, who it is hard to come close while he is in his palace. However, when the king goes out of the palace, everyone comes to see him. All the people call out to him and he hears everyone. Lehavdil, when the Divine Presence is in exile, so to speak, during the 3 weeks of mourning for the Beis Hamikdash, everyone has the ability to come close and call out to the King of the world. Then when it comes to the days of Elul and Tishrei, and the King is back in the palace, one already has a relationship with Him. This is the unbelievable positive lesson learnt out from the very beginning of Megilas Eicha.
Many think of this time period as just restrictions on music and other activities, but the Magid of Mezeritch is teaching us that it is so much more than that. It is such a special time to come closer to Hashem. One way would be to say Tikun Chatzos at least once in a while to show we are at least a little concerned about the loss of the King. Why wait till Elul or Aseres Yemei Teshuva to do tershuva, when we have an opportunity right now to get a head start on things in a time when we can get closer to Hashem then later on. Whether it's learning more Torah, improving davening, doing more chesed or any other Avodas Hashem, this is a special opportunity that should be taken advantage of by all. May we merit truly coming close to the King of kings and seeing the rebuilding of His palace speedily in our days.