Moses' wife was of another race and in the Bible (Numbers 12:1-15) Aaron and Miriam were punished for criticizing this interracial marriage. What does the Qur'an say about interracial marriages?
In Quran, Allah said:
يَا أَيُّهَا النَّاسُ إِنَّا خَلَقْنَاكُم مِّن ذَكَرٍ وَأُنثَى وَجَعَلْنَاكُمْ شُعُوبًا وَقَبَائِلَ لِتَعَارَفُوا إِنَّ أَكْرَمَكُمْ عِندَ اللَّهِ أَتْقَاكُمْ إِنَّ اللَّهَ عَلِيمٌ خَبِيرٌ
O mankind, indeed We have created you from male and female and made you peoples and tribes that you may know one another. Indeed, the most noble of you in the sight of Allah is the most righteous of you. Indeed, Allah is Knowing and Acquainted. [Quran 49:13]
Allah clearly encourages us to knowing one another as stated in the verse. One of the ways of knowing other people and races is marriage.
Also, Prophet Mohammed (PBUH) Married to different women from different races and tribes and religions. Maria (RA) was Coptic (Native Egyptian) and she was Christian (Converted to Islam later), Safia (RA) was Jewish from Israelites (Also converted to Islam).
As long as Allah encourages knowing other races and Mohammed (PBUH) married to another races, then it is something actually not only allowed but also recommended for Muslims.
in sura Al-Hujurat:13
يَا أَيُّهَا النَّاسُ إِنَّا خَلَقْنَاكُم مِّن ذَكَرٍ وَأُنثَىٰ وَجَعَلْنَاكُمْ شُعُوبًا وَقَبَائِلَ لِتَعَارَفُوا إِنَّ أَكْرَمَكُمْ عِندَ اللَّـهِ أَتْقَاكُمْ إِنَّ اللَّـهَ عَلِيمٌ خَبِيرٌ
O men, We created you from a male and female, and formed you into nations and tribes that you may recognize each other. He who has more integrity has indeed greater honor with God. Surely God is all-knowing and well-informed.
This indicates that it is not preferred between one interracial over another only piety and belief And Islam encourage acquaintance and love between the interracial
Of course, with the condition of the same religion Depending on the Al-Baqara:221
وَلَا تَنكِحُوا الْمُشْرِكَاتِ حَتَّىٰ يُؤْمِنَّ ۚ وَلَأَمَةٌ مُّؤْمِنَةٌ خَيْرٌ مِّن مُّشْرِكَةٍ وَلَوْ أَعْجَبَتْكُمْ ۗ وَلَا تُنكِحُوا الْمُشْرِكِينَ حَتَّىٰ يُؤْمِنُوا ۚ وَلَعَبْدٌ مُّؤْمِنٌ خَيْرٌ مِّن مُّشْرِكٍ وَلَوْ أَعْجَبَكُمْ ۗ أُولَـٰئِكَ يَدْعُونَ إِلَى النَّارِ ۖ وَاللَّـهُ يَدْعُو إِلَى الْجَنَّةِ وَالْمَغْفِرَةِ بِإِذْنِهِ ۖ وَيُبَيِّنُ آيَاتِهِ لِلنَّاسِ لَعَلَّهُمْ يَتَذَكَّرُونَ
Do not marry idolatrous women unless they join the faith. A maid servant who is a believer is better than an idolatress even though you may like her. And do not marry your daughters to idolaters until they accept the faith. A servant who is a believer is better than an idolater even though you may like him. They invite you to Hell, but God calls you to Paradise and pardon by His grace. And He makes His signs manifest that men may haply take heed.
Knowing that the islam (sharee'ah) has Allowed to Muslim men marriage from non-Muslim women, but not vice versa.
I think your question has been answered already but just a side note - the story of Bilal RA is a beautiful story, how he began as a slave and ended up being one of the most trusted and loyal of companions to the Prophet pbuh as a man of Abyssinian (Ethiopian) decent. He also went on to become the first Muaddhin (caller to prayer) in Islam. To me, giving such a man such status shows how Islam tried to eradicate racism and show us that no man is better than another man except in piety.