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Actually, 5 of the 10 Commandments can already be considered "secular", which is to say, not an instruction on worship, but of common, civilized behavior. Let's review, and "tweak" to avoid offending folks of various religious and/or non religious leanings:

1.Thou shalt have no other gods before me.

OK, this one is clearly religious, and even among believers in God there is much disagreement about what it means to be "God". If you want to "secularize" this, you are going to end up with something VERY CLOSE to the US Constitution 1st Amendment. Something like this:
Worship YOUR God in YOUR Way, but DO NOT INTERFERE with others as they Worship THEIR God in THEIR Way.

2.Thou shalt not make unto thee any graven image, or any likeness of any thing that is in heaven above, or that is in the earth beneath, or that is in the water under the earth. Thou shalt not bow down thyself to them, nor serve them: for I the Lord thy God am a jealous God, visiting the iniquity of the fathers upon the children unto the third and fourth generation of them that hate me; And shewing mercy unto thousands of them that love me, and keep my commandments.

Alrighty, this one is also specifically religious. I think that it might be covered in the fairly broad suggestion offered for (1) above, so I will repeat it:
Worship YOUR God in YOUR Way, but DO NOT INTERFERE with others as they Worship THEIR God in THEIR Way.

3.Thou shalt not take the name of the Lord thy God in vain; for the Lord will not hold him guiltless that taketh his name in vain.

Another clearly religious commandment, but one I think can be handled with an admonition to be "respectful" and "polite". Something like this:
Just as YOU would find it offensive to have your religion Mocked, refrain from speech and actions that mock and belittle the religions of others.

4.Remember the sabbath day, to keep it holy. Six days shalt thou labour, and do all thy work: But the seventh day is the sabbath of the Lord thy God: in it thou shalt not do any work, thou, nor thy son, nor thy daughter, thy manservant, nor thy maidservant, nor thy cattle, nor thy stranger that is within thy gates: For in six days the Lord made heaven and earth, the sea, and all that in them is, and rested the seventh day: wherefore the Lord blessed the sabbath day, and hallowed it.

Already covered in the fairly broad suggestion offered for (1) above, so I will repeat it:
Worship YOUR God in YOUR Way, but DO NOT INTERFERE with others as they Worship THEIR God in THEIR Way.

5.Honour thy father and thy mother: that thy days may be long upon the land which the Lord thy God giveth thee.

Common Decency codified:
Respect your elders, for they have lived and learned beyond your years, and in their declining years, will need your help and love.

6.Thou shalt not kill.

Already codified as law in virtually every civilized culture:
Don't Murder People, that's really bad.

7.Thou shalt not commit adultery.

Codified as law in many civilized cultures:
Don't Violate the Sanctity of Marriage, it leads to really bad outcomes.

8.Thou shalt not steal.

Already codified as law in virtually every civilized culture:
Don't Steal things from people, that's really bad.

9.Thou shalt not bear false witness against thy neighbour.

Already codified as law in virtually every civilized culture with respect to trials:
Don't Lie To People, that's really bad.

10.Thou shalt not covet thy neighbour’s house, thou shalt not covet thy neighbour’s wife, nor his manservant, nor his maidservant, nor his ox, nor his ass, nor any thing that is thy neighbour’s.

Not a law, but thought of as basic common sense in virtually every civilized culture:
Don't be jealous of what other people have. Earn your own if you want what they have.

See? There is a calm, thought out response.

BTW, can anyone name a religion that routinely violates any of the “suggestions” I offered? I can think of one off the top of my head...

You sure you don’t have the source in reverse, assuming that people thought of these mores on their own? Take note of Romans 2:14.

I didn’t labor under the assumption that worrying about offending this or that person or group was something that FR burdened itself with, but rather upholding the principles of the US republic no matter who takes offense.