If these restaurants are in a country where the slaughter of meat is taken care of by Muslims or People of the Book (i.e., Jews and Christians), or the restaurant owners take care of the slaughter themselves and they are Muslims or people of the Book, then it is permissible to eat there, even if we do not know whether they mentioned the name of Allaah over the slaughter or not, because the basic principle is that their meat is allowed. Al-Bukhaari (2057) narrated from ‘Aa’ishah (may Allaah be pleased with her) that some people said, “O Messenger of Allaah, some people bring meat to us and we do not know whether they mentioned the name of Allaah over it or not.” The Messenger of Allaah (peace and blessings of Allaah be upon him) said: “Say the name of Allaah over it and eat it.” 

But if the slaughter of meat is taken care of by people other than Muslims or people of the Book, such as atheists and Hindus, then it is not permissible to eat it. 

It should be noted that meat slaughtered by a Muslim or a Jew or Christian is permissible if it was slaughtered in the manner prescribed in sharee’ah or if we do not know how it was slaughtered. But if we know that it was slaughtered in a way other than that prescribed in sharee’ah, such as by strangling, electric shock, etc, then it is maytah or dead meat and it is haraam to eat it regardless of whether the one who killed it was a Muslim or a kaafir, because Allaah says (interpretation of the meaning): 

“Forbidden to you (for food) are: Al‑Maitah (the dead animals — cattle — beast not slaughtered), blood, the flesh of swine, and that on which Allaah’s Name has not been mentioned while slaughtering (that which has been slaughtered as a sacrifice for others than Allaah, or has been slaughtered for idols) and that which has been killed by strangling, or by a violent blow, or by a headlong fall, or by the goring of horns — and that which has been (partly) eaten by a wild animal — unless you are able to slaughter it (before its death)”

[al-Maa'idah 5:3]

Shaykh Ibn Baaz in Fataawa Islamiyyah, 3/414

Shaykh Ibn ‘Uthaymeen said: 

One of the following three scenarios must apply to this meat: 

1 – We know that it has been slaughtered in the proper manner. This meat is halaal. 

2 – We know that it has not been slaughtered in the proper manner. This meat is haraam. 

3 – We are not sure; we do not know whether it has been slaughtered in the proper manner or not. The ruling in this case is that the meat is halaal, and we do not have to ask about how it was killed, or whether the name of Allaah was mentioned over it or not. Rather it seems from the Sunnah that it is better not to ask or try to find out. Hence when they said to the Prophet (peace and blessings of Allaah be upon him), “we do not know whether they mentioned the name of Allaah over it or not,” he did not say to them, Ask them whether they mentioned the name of Allaah over them or not, rather he said: “Say the name of Allaah over it yourselves, and eat it.” This tasmiyah or saying of the name of Allaah which the Prophet (peace and blessings of Allaah be upon him) told them to do is not the tasmiyah of slaughter, because the slaughter is over and done with. Rather it is the tasmiyah of eating, because what is prescribed is for the person who is eating to say the name of Allaah when he eats. The correct view regarding saying the name of Allaah when eating is that it is obligatory, because the Messenger of Allaah (peace and blessings of Allaah be upon him) enjoined it, and because if a person does not say the name of Allaah, the Shaytaan shares his food and drink. 

If a person wants to be cautious and forego eating this meat, there is nothing wrong with that, but there is also nothing wrong with eating it. 

From Fataawa Islamiyyah, 3/415 

And Allaah knows best.